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Automated vehicles (AVs) are expected to be beneficial for Mobility-on-Demand (MoD), thanks to their ability of being globally coordinated. To facilitate the steady transition towards full autonomy, we consider the transition period of AV deployment, whereby an MoD system operates a mixed fleet of automated vehicles (AVs) and human-driven vehicles (HVs). In such systems, AVs are centrally coordinated by the operator, and the HVs might strategically respond to the coordination of AVs. We devise computationally tractable strategies to coordinate mixed fleets in MoD systems. Specifically, we model an MoD system with a mixed fleet using a Stackelberg framework where the MoD operator serves as the leader and human-driven vehicles serve as the followers. We develop two models: 1) a steady-state model to analyze the properties of the problem and determine the planning variables (e.g., compensations, prices, and the fleet size of AVs), and 2) a time-varying model to design a real-time coordination algorithm for AVs. The proposed models are validated using a case study inspired by real operational data of a MoD service in Singapore. Results show that the proposed algorithms can significantly improve system performance.